1. Source-Free Series
RLC Circuits

2. Objective of Lecture
Derive the equations that relate the voltages across and currents flowing through a resistor, an inductor, and a capacitor in series as:
the unit step function associated with voltage or current source changes from 1 to 0 or
a switch disconnects a voltage or current source into the circuit.
Describe the solution to the 2nd order equations when the condition is:
Overdamped
Critically Damped
Underdamped

3. Series RLC Network
With a step function voltage source.

4. Boundary Conditions
You must determine the initial condition of the inductor and capacitor at t < to and then find the final conditions at t = ∞s.
Since the voltage source has a magnitude of 0V at t < to
i(to-) = iL(to-) = 0A and vC(to-) = Vs
vL(to-) = 0V and iC(to-) = 0A
Once the steady state is reached after the voltage source has a magnitude of Vs at t > to, replace the capacitor with an open circuit and the inductor with a short circuit.
i(∞s) = iL(∞s) = 0A and vC(∞s) = 0V
vL(∞s) = 0V and iC(∞s) = 0A

5. Selection of Parameter
Initial Conditions
i(to-) = iL(to-) = 0A and vC(to-) = Vs
vL(to-) = 0V and iC(to-) = 0A
Final Conditions
i(∞s) = iL(∞s) = 0A and vC(∞s) = oV
vL(∞s) = 0V and iC(∞s) = 0A
Since the voltage across the capacitor is the only parameter that has a non-zero boundary condition, the first set of solutions will be for vC(t).

6. Kirchhoff’s Voltage Law

7. Let vC(t) = AesDt where Dt=t-to
General Solution
Let vC(t) = AesDt where Dt=t-to

8. General Solution (con’t)

9. General Solution (con’t)

10. General Solution (con’t)

11. Solve for Coefficients A1 and A2
Use the boundary conditions at to- and t = ∞s to solve for A1 and A2.
Since the voltage across a capacitor must be a continuous function of time.
Also know that

12. Overdamped Case a > wo implies that C > 4L/R2
s1 and s2 are negative and real numbers

13. Critically Damped Case
a = wo
implies that C = 4L/R2
s1 = s2 = - a = -R/2L

14. Underdamped Case a < wo implies that C < 4L/R2
, i is used by the mathematicians for imaginary numbers

16. Angular Frequencies wo is called the undamped natural frequency
The frequency at which the energy stored in the capacitor flows to the inductor and then flows back to the capacitor. If R = 0W, this will occur forever.
wd is called the damped natural frequency
Since the resistance of R is not usually equal to zero, some energy will be dissipated through the resistor as energy is transferred between the inductor and capacitor.
a determined the rate of the damping response.

18. Properties of RLC network
Behavior of RLC network is described as damping, which is a gradual loss of the initial stored energy
The resistor R causes the loss
a determined the rate of the damping response
If R = 0, the circuit is loss-less and energy is shifted back and forth between the inductor and capacitor forever at the natural frequency.
Oscillatory response of a lossy RLC network is possible because the energy in the inductor and capacitor can be transferred from one component to the other.
Underdamped response is a damped oscillation, which is called ringing.

19. Properties of RLC network
Critically damped circuits reach the final steady state in the shortest amount of time as compared to overdamped and underdamped circuits.
However, the initial change of an overdamped or underdamped circuit may be greater than that obtained using a critically damped circuit.

20. Set of Solutions when t > to
There are three different solutions which depend on the magnitudes of the coefficients of the and the terms.
To determine which one to use, you need to calculate the natural angular frequency of the series RLC network and the term a.

21. Transient Solutions when t > to
Overdamped response (a > wo)
Critically damped response (a = wo)
Underdamped response (a < wo)

22. Find Coefficients
After you have selected the form for the solution based upon the values of wo and a
Solve for the coefficients in the equation by evaluating the equation and its first derivate at t = to- using the initial boundary conditions.
vC(to-) = Vs and dvC(to-)/dt = iC (to-)/C = 0V/s

23. Other Voltages and Currents
Once the voltage across the capacitor is known, the following equations for the case where t > to can be used to find:

24. Solutions when t < to
The initial conditions of all of the components are the solutions for all times -∞s < t < to.
vC(t) = Vs
iC(t) = 0A
vL(t) = 0V
iL(t) = 0A
vR(t) = 0V
iR(t) = 0A

25. Summary
The set of solutions when t > to for the voltage across the capacitor in a RLC network in series was obtained.
Selection of equations is determine by comparing the natural frequency wo to a.
Coefficients are found by evaluating the equation and its first derivation at t = to-.
The voltage across the capacitor is equal to the initial condition when t < to
Using the relationships between current and voltage, the current through the capacitor and the voltages and currents for the inductor and resistor can be calculated.